Transversality and the Inverse Image of a Submanifold with Corners

By Lars Tyge Nielsen

Mathematica Scandinavica 49 (1981), 211-221

Introduction

It is a well-known theorem about usual differential manifolds that the inverse image of a submanifold by a map which intersects the submanifold transversally is a submanifold of the same codimension. This paper generalizes the theorem to the case where the manifolds are allowed to have corners (Section 6). The key step in the proof is a generalization of the theorem about local linearization of submersions (Section 5). It is assumed that the map preserves local facets relatively to the submanifold. This property is defined in Section 4. It is convenient to make extensive use of germs, and therefore some notation related to germs is introduced in Section 2. Finally, the special case of manifolds and submanifolds with boundary of codimension one is discussed in Section 7.